MinT - Best Known (64, 64+41, s)-Nets in Base 25

Best Known (64, 64+41, s)-Nets in Base 25

(64, 64+41, 378)-Net over F₂₅ — Constructive and digital

Digital (64, 105, 378)-net over F₂₅, using

25¹ times duplication [i] based on digital (63, 104, 378)-net over F₂₅, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (10, 23, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]
  2. digital (10, 30, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)
  3. digital (10, 51, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(64, 64+41, 3090)-Net over F₂₅ — Digital

Digital (64, 105, 3090)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+41, 6432537)-Net in Base 25 — Upper bound on s

There is no (64, 105, 6432538)-net in base 25, because

1 times m-reduction [i] would yield (64, 104, 6432538)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 24 308680 183996 779334 754789 197160 283424 946229 923194 983070 793049 717320 380091 958124 369161 985916 611423 107139 905242 020124 153133 681279 305478 167082 763585 > 25¹⁰⁴ [i]